From Relativities to a Complete Table
A rating plan may describe a risk using two characteristics, such as territory and vehicle type. When each characteristic has several levels, calculating one risk's rate at a time makes it easy to omit a combination or use the wrong factor. A two-way rating table organizes every combination in one place: rows represent levels of one characteristic, columns represent levels of the other, and each cell contains the indicated rate for that combination.
The construction uses the multiplicative rating structure established in the earlier material on rating cells, base rates, reference levels, and individual indicated rates. The task here is to apply that structure consistently across a complete table and check that its entries agree with the rating formula.
A table is not just a display of numbers. It records the rate implied by every pairing of the two factor levels. The row and column headings should therefore name the characteristics and their levels, and the entries should be rates—not relativities—unless the table is explicitly labelled otherwise.
Why the Cell Formula Works
Each relativity expresses a multiplicative adjustment relative to its own reference level. Holding the second characteristic at reference, changing the first from reference level to level \(i\) changes the base rate from \(B\) to \(B r_i\). Changing the second characteristic to level \(j\) then applies its factor \(s_j\) to that amount. Thus the combined rate is \(B r_i s_j\). Equivalently, the ratio of the cell rate to the base rate is the product of both relativities:
This calculation assumes the rating plan specifies separate multiplicative factors and no additional cell-specific adjustment. It does not assert that the two characteristics are statistically independent; it describes how the stated rating formula combines their selected factors. If a plan includes an interaction or a separate adjustment for a particular pairing, that term must also be included. The table built here is for the two-factor multiplicative formula as stated.
Put the levels of the first characteristic in rows and the levels of the second in columns. Keep the factor values clearly associated with their levels.
For every cell, multiply the base rate by the row relativity and the column relativity.
The cell with both reference levels must equal the base rate. Recalculate other cells directly from their two factors.
Carry full precision through the multiplication, then round displayed money to the nearest cent when rates are reported in dollars.
Worked Tables
Worked Example: Completing a Three-by-Three Table
A hypothetical insurer uses a base rate of $240 per vehicle for a specified coverage and policy term. Territory relativities are 0.80 for Rural, 1.00 for Standard, and 1.25 for Urban. Vehicle-use relativities are 0.90 for Occasional, 1.10 for Regular, and 1.30 for Frequent. Construct the full indicated-rate table. This is deterministic rating arithmetic; no interest, mortality, or loss-distribution assumption is needed.
Method. Multiply $240 by the territory relativity and the vehicle-use relativity for each cell. Keep the full product until the final rate is stated.
For Rural–Occasional, the calculation is \(240(0.80)(0.90)=240(0.72)=\$172.80\). For Rural–Regular, it is \(240(0.80)(1.10)=240(0.88)=\$211.20\). For Rural–Frequent, it is \(240(0.80)(1.30)=240(1.04)=\$249.60\).
For Standard–Occasional, \(240(1.00)(0.90)=\$216.00\). For Standard–Regular, \(240(1.00)(1.10)=\$264.00\). For Standard–Frequent, \(240(1.00)(1.30)=\$312.00\).
For Urban–Occasional, \(240(1.25)(0.90)=240(1.125)=\$270.00\). For Urban–Regular, \(240(1.25)(1.10)=240(1.375)=\$330.00\). For Urban–Frequent, \(240(1.25)(1.30)=240(1.625)=\$390.00\).
| Territory / Vehicle use | Occasional (0.90) | Regular (1.10) | Frequent (1.30) |
|---|---|---|---|
| Rural (0.80) | $172.80 | $211.20 | $249.60 |
| Standard (1.00) | $216.00 | $264.00 | $312.00 |
| Urban (1.25) | $270.00 | $330.00 | $390.00 |
Check. Standard–Regular is the reference combination, so its rate must equal the $240 base rate only if Regular is the vehicle-use reference. Here Regular has relativity 1.10, not 1.00; the stated factor set has no reference vehicle-use level. Therefore the table's $264.00 is consistent with the supplied factors, but the base rate is not itself a cell in this table. This is an important check: a stated base rate is the reference-risk rate, and it appears as a cell only when both selected relativities are 1.00.
Mistake avoided. Treating the $240 base rate as the rate for Standard–Regular would ignore the 1.10 vehicle-use factor. The correct calculation includes both factors and gives $264.00.
Worked Example: Constructing a Table With a Reference Cell
A hypothetical insurer has a base rate of $315 per unit of coverage. The reference construction type has relativity 1.00, while the other type has relativity 0.80. For annual mileage, the levels Low, Medium, and High have relativities 0.95, 1.05, and 1.40. Build the complete table, round rates to cents, and verify the reference cell. No interest, mortality, or severity-distribution assumption is used.
Method. Use the construction type as the row factor and mileage as the column factor. The two reference levels are Reference construction (1.00) and Medium mileage (1.05) only if Medium is designated as the reference; however, its supplied relativity is not 1.00. Thus the base rate is not a cell in the table as given. The calculation still follows the specified factors.
For the 0.80 construction type, the Low, Medium, and High cells are respectively \(315(0.80)(0.95)=315(0.76)=\$239.40\), \(315(0.80)(1.05)=315(0.84)=\$264.60\), and \(315(0.80)(1.40)=315(1.12)=\$352.80\).
For Reference construction, the three cells are \(315(1.00)(0.95)=\$299.25\), \(315(1.00)(1.05)=\$330.75\), and \(315(1.00)(1.40)=\$441.00\).
| Construction / Annual mileage | Low (0.95) | Medium (1.05) | High (1.40) |
|---|---|---|---|
| Other (0.80) | $239.40 | $264.60 | $352.80 |
| Reference construction (1.00) | $299.25 | $330.75 | $441.00 |
Check. In the reference-construction row, the rates equal the base rate multiplied by each mileage relativity. For example, \(299.25/315=0.95\), and \(441.00/315=1.40\). There is no mileage level with relativity 1.00, so no cell equals the base rate. If Medium were intended to be the reference mileage level, its factor would need to be 1.00, or the supplied base rate would need to be defined for a different reference combination.
Mistake avoided. Do not label a level “reference” merely because it sounds typical. The factor value, not the label, determines whether it is a reference level.
Worked Example: Auditing a Complete Table
For a hypothetical specialty vehicle program, the base rate is $425. Age-band relativities are 0.70, 1.00, and 1.35; annual-use relativities are 0.85, 1.00, and 1.20. Construct the table, then audit the cell for the highest age band and lowest annual use. Report dollars to cents. There are no interest, mortality, or distribution assumptions in this calculation.
Method. Multiply the base rate by the factor for each row and column. The middle levels both have relativity 1.00, so their intersection should reproduce the base rate.
For the first age band (0.70), the three rates are \(425(0.70)(0.85)=425(0.595)=\$252.875\), \(425(0.70)(1.00)=\$297.50\), and \(425(0.70)(1.20)=\$357.00\). Rounding the first to cents gives $252.88.
For the middle age band (1.00), the rates are \(425(1.00)(0.85)=\$361.25\), \(425(1.00)(1.00)=\$425.00\), and \(425(1.00)(1.20)=\$510.00\).
For the highest age band (1.35), the rates are \(425(1.35)(0.85)=425(1.1475)=\$487.6875\), \(425(1.35)(1.00)=\$573.75\), and \(425(1.35)(1.20)=425(1.62)=\$688.50\). Thus the audited highest-age, lowest-use rate is $487.69.
| Age band / Annual use | Low (0.85) | Middle (1.00) | High (1.20) |
|---|---|---|---|
| First (0.70) | $252.88 | $297.50 | $357.00 |
| Middle (1.00) | $361.25 | $425.00 | $510.00 |
| Highest (1.35) | $487.69 | $573.75 | $688.50 |
Check. The middle-middle cell is $425.00, matching the base rate because both factors equal 1.00. In the highest-age row, the ratio of the high-use rate to the middle-use rate is \(688.50/573.75=1.20\), which matches the high-use relativity. The independently calculated low-use cell also agrees with its table entry after rounding.
Mistake avoided. Rounding the intermediate product 1.1475 or the unrounded dollar result too early can distort later checks. Retain precision through multiplication and round only the final displayed rate.
Checks That Catch Table Errors
A few systematic checks are more reliable than inspecting a table by eye. First, confirm that the table dimensions match the number of levels in each factor set. Second, recalculate the reference combination, if present. Third, select cells from different rows and columns and apply \(R_{ij}=B r_i s_j\) directly. Finally, compare rates within a row or column: holding one characteristic fixed, changing the other should change the rate in proportion to the corresponding relativity ratio.
For example, with a fixed row factor \(r_i\), two column rates satisfy
The base rate and fixed row factor cancel. This gives an independent check on the pattern of a completed table without recalculating every cell from scratch.
A frequent exam trap is to add percentage changes instead of applying the stated factors multiplicatively. Suppose the base rate is $240, and two factors are 1.20 and 1.10. The incorrect additive approach treats these as increases of 20% and 10%, yielding \(240(1+0.20+0.10)=\$312\). The rating formula instead gives \(240(1.20)(1.10)=240(1.32)=\$316.80\). The difference arises because the second multiplicative factor applies to the already adjusted amount. Use the formula and factor values as stated; do not substitute a sum of percentage changes.
Building the table makes the rating plan’s implications visible across all combinations. It also helps identify a missing level, a mislabeled reference, a misplaced factor, or a cell inconsistent with the formula. When rates are reported in dollars, keep unrounded products during the calculations and round the final entries to cents.
Check Your Understanding
For each problem, use the indicated multiplicative structure and check the result against the stated factors.
- A hypothetical plan has a base rate of $280, with a row relativity of 0.90 and a column relativity of 1.25. What is the indicated cell rate? (Answer: $315.00)
- A base rate is $460. One factor is 1.08 and the other is 0.85. Calculate the cell rate to cents. (Answer: $422.28)
- A plan has base rate $200, row relativities 0.75 and 1.00, and column relativities 0.90 and 1.20. Give the four-cell table in row order. (Answer: First row: $135.00, $180.00; second row: $180.00, $240.00)
- A base rate is $350. Two levels have relativities 1.15 and 0.80. An analyst reports a combined rate of $332.50 by adding the percentage changes. Calculate the correct rate and state the source of the error. (Answer: $322.00; the relativities should be multiplied, so \(350(1.15)(0.80)=350(0.92)\), not computed by adding the percentage changes.)
- A two-factor table has base rate $500. In one fixed row, the rate at column factor 0.90 is $360. What should the rate be at column factor 1.15, with the same row factor? (Answer: $460; the ratio is \(1.15/0.90\), so \(360(1.15/0.90)=460\).)